Primes in intervals of bounded length

The infamous twin prime conjecture states that there are infinitely many pairs of distinct primes which differ by $2$. Until recently this conjecture had seemed to be far out of reach with current techniques. However, in April 2013, Yitang Zhang proved the existence of a finite bound $B$ such that there are infinitely many pairs of distinct primes which differ by no more than $B$. This is a massive breakthrough, making the twin prime conjecture look highly plausible, and the techniques developed help us to better understand other delicate questions about prime numbers that had previously seemed intractable.

Zhang even showed that one can take $B = 70000000$. Moreover, a co-operative team, Polymath8, collaborating only online, had been able to lower the value of $B$ to ${4680}$. They had not only been more careful in several difficult arguments in Zhang's original paper, they had also developed Zhang's techniques to be both more powerful and to allow a much simpler proof (and this forms the basis for the proof presented herein).

In November 2013, inspired by Zhang's extraordinary breakthrough, James Maynard dramatically slashed this bound to $600$, by a substantially easier method. Both Maynard and Terry Tao, who had independently developed the same idea, were able to extend their proofs to show that for any given integer $m\geq 1$ there exists a bound $B_m$ such that there are infinitely many intervals of length $B_m$ containing at least $m$ distinct primes. We will also prove this much stronger result herein, even showing that one can take $B_m=e^{8m+5}$.

If Zhang's method is combined with the Maynard-Tao setup, then it appears that the bound can be further reduced to $246$. If all of these techniques could be pushed to their limit, then we would obtain $B$($=B_2$)$=12$ (or arguably to $6$), so new ideas are still needed to have a feasible plan for proving the twin prime conjecture.

The article will be split into two parts. The first half will introduce the work of Zhang, Polymath8, Maynard and Tao, and explain their arguments that allow them to prove their spectacular results. The second half of this article develops a proof of Zhang's main novel contribution, an estimate for primes in relatively short arithmetic progressions.